New scaling laws are presented for the spatial variation of the mean v
elocity and lateral extent of a two-dimensional turbulent wall jet, fl
owing over a fixed rough boundary. These scalings are analogous to tho
se derived by Wygnanski et nl. (1992) for the flow of a wall jet over
a smooth boundary. They reveal that the characteristics of the jet dep
end weakly upon the roughness length associated with the boundary, as
confirmed by experimental studies (Rajaratnam 1967). These laws are us
ed in the development of an analytical framework to model the progress
ive erosion of an initially flat bed of grains by a turbulent jet. The
grains are eroded if the shear stress, exerted on the grains at the s
urface of the bed, exceeds a critical value which is a function of the
physical characteristics of the grains. After the walljet has been fl
owing for a sufficiently long period, the boundary attains a steady st
ate, in which the mobilizing forces associated with the jet are insuff
icient to further erode the boundary. The steady-state profile is calc
ulated separately by applying critical conditions along the bed surfac
e for the incipient motion of particles. These conditions invoke a rel
ationship between the mobilizing force exerted by the jet, the weight
of the particles and the local gradient of the bed. Use of the new sca
ling laws for the downstream variation of the boundary shear stress th
en permits the calculation of the shape of the steady-state scour pit.
The predicted profiles are in good agreement with the experimental st
udies on the erosive action of submerged water and air jets on beds of
sand and polystryene particles (Rajaratnam 1981). The shape of the er
oded boundary at intermediate times, before the steady state is attain
ed, is elucidated by the application of a sediment-volume conservation
equation. This relationship balances the rate of change of the bed el
evation with the divergence of the flux of particles in motion. The fl
ux of particles in motion is given by a semiempirical function of the
amount by which the boundary shear stress exceeds that required for in
cipient motion. Hence the conservation equation may be integrated to r
eveal the transient profiles of the eroded bed. There is good agreemen
t between these calculated profiles and experimental observations (Raj
aratnam 1981).